Fuel-cell parameter estimation based on improved gorilla troops technique

The parameter extraction of the proton exchange membrane fuel cells (PEMFCs) is an active study area over the past few years to achieve accurate current/voltage (I/V) curves. This work proposes an advanced version of an improved gorilla troops technique (IGTT) to precisely estimate the PEMFC’s model parameters. The GTT's dual implementation of the migration approach enables boosting the exploitation phase and preventing becoming trapped in the local minima. Besides, a Tangent Flight Strategy (TFS) is incorporated with the exploitation stage for efficiently searching the search space. Using two common PEMFCs stacks of BCS 500W, and Modular SR-12, the developed IGTT is effectively applied. Furthermore, the two models are evaluated under varied partial temperature and pressure. In addition to this, different new recently inspired optimizers are employed for comparative validations namely supply demand optimization (SDO), flying foxes optimizer (FFO) and red fox optimizer (RFO). Also, a comparative assessment of the developed IGTT and the original GTT are tested to ten unconstrained benchmark functions following to the Congress on Evolutionary Computation (CEC) 2017. The proposed IGTT outperforms the standard GTT, grey wolf algorithm (GWA) and Particle swarm optimizer (PSO) in 92.5%, 87.5% and 92.5% of the statistical indices. Moreover, the viability of the IGTT is proved in comparison to various previously published frameworks-based parameter's identification of PEMFCs stacks. The obtained sum of squared errors (SSE) and the standard deviations (STD) are among the difficult approaches in this context and are quite competitive. For the PEMFCs stacks being studied, the developed IGTT achieves exceedingly small SSE values of 0.0117 and 0.000142 for BCS 500 and SR-12, respectively. Added to that, the IGTT gives superior performance compared to GTT, SDO, FFO and RFO obtaining the smallest SSE objective with the least STD ever.

Mathematical model of PEMFCs. To form the PEMFC's model, the I-V characteristics (Polarization curves) could be mathematically displayed. The steady state performance of the PEMFCs is described using the electrochemical simplified model presented by Mann et al. 11 . This paradigm is widely employed in numerous literature analyses. The mathematical model for the stack's output voltage ( V stack ) as illustrated in (4), which comprises of several series-connected cells ( N cells ) 11,19,41,42 .
whereas v act refers to the cell activation overpotential, E Nernst is the Nernst voltage per cell, v conc indicates the concentration over-potential, and v refers to the cell ohmic drop in voltage. The voltage E Nernst can be calculated using Eq. (2) under a reference temperature of 25 °C. Thus, these three voltages drop are provided as depicted in Eqs. (5)-(8) 11,41 .
where P O2 and P H2 illustrate the regulating pressures of oxygen (O 2 ) (atm) and hydrogen (H 2 ), respectively, while T fc represents the working temperature of the FC (K). Moreover, C O2 manifests the concentration of O 2 (mol/ cm 3 ), M A signifies the membrane area (cm), whereas I fc is its current (A) and ξ 1 − ξ 4 characterizes semiempirical coefficients 3,41 . Besides, l indicates the thickness of membrane (cm), whilst R c and R m reveal the leads and the membrane ohmic resistances (Ω); respectively. In addition to this, ρ m demonstrates the membrane resistivity (Ω.cm), β is handled as an empirical constant, and λ is treated as a changeable parameter, whilst J max and J describe the maximum and actual thermal current densities (A/cm 2 ), respectively 3,19 . www.nature.com/scientificreports/ where F, ℘ and α represent Faraday's, ideal gas constants, and charge transfer coefficient, respectively. It becomes clear that the temperature and current density can affect the concentrating voltage drop in linear relation based on a thorough understanding of Eqs. (5) and (6). To illustrate, at higher cell temperatures and larger current densities, the concentration polarization voltage is projected to be increased 11,41 . It can be deduced that the seven parameters are fundamentally approximated to construct an appropriate PEMFC's model.

Proposed methodology
Problem formulation and description. Due to a lack of manufacturer data, the PEMFC's modelling has significant non-linear characteristics and various undetermined parameters. This signifies that creating an accurate model will be incredibly challenging. Seven parameters should be calculated. The optimization objective (FCF), in this purpose, is stated as the minimization of the sum squared error between the experimental FC voltage and estimated model voltage. Thus, the PEMFCs parameters estimation problem is approached as an objective target. The issue in the present work may be thought of as a non-convex optimization issue. The FCF is written in Eq. (10) as follows 19,43 .
where m expresses the iteration counter, N samples designates the number of measured voltage data, V FC,est manifests the FC estimated calculated voltage, and V FC,exp represents the measured output voltage of the model. The optimization objective is restricted with inequality constraints for unknown seven parameters which are the minimum and maximum limits of these parameters. The GTT is applied to optimize these seven unknown parameters which are namely, λ, ξ 1 − ξ 4 , R c , and β that obtain the best value of the SSE.
Gorilla troops technique. The GTT depends on several distinct behaviors of the gorillas that are mathematically simulated. Five behaviors are taken into account in this situation to optimize gorilla behavior: three for the exploration stage and two for the exploitation stage. These activities include migration to a strange region, traveling to other gorillas, migration toward a specified spot, competing for adult females and escorting the silverback. Two stages represent these strategic options that can be divided into the exploitation stage and exploration stage as will be manifested in the following subsections.
Exploration stage. Three distinct behaviors, in this stage, are elaborated: the first one is to manifest GTT exploration (which is movement to an unidentified end point), whereas the second tactic represents the traveling behavior to other gorillas. Furthermore, the third tactic aims at encouraging GTT competences in determining a myriad of calculation spaces that represents the migration toward a specified spot. Equation (11) can represent these three behaviors mathematically, where the movement to unidentified end point tactic, in this equation, is selected if a random number (rn) is smaller than a factor (Fr). Besides, the traveling to other gorillas or migration toward a specified is carefully selected if a random number equals/ (is more than) 50%.
where rn, rn 1 , rn 2 , rn 3 , and rn 4 illustrate random values among [0, 1], whilst X(Itn) and GtX(Itn + 1) define the full and forthcoming vectors of the gorilla's position. The arbitrary assignable variables X r and GtXr could be used to ascertain a gorilla's current group and potential position. The factor (Fr) is, in the range [0:1] and characterizes the possibility of deciding on a migrating method to an unsettled location. The LB and UB are the variables ' minimum and maximum bounds. the variables D and Q could be determined mathematically by Eqs. Exploitation stage. Two tactics in this stage are proposed when the factor D × (1 − Itn/MxItn) is compared with the variable (Y). These two behaviors are the escorting the silverback and the competing for adult females. The first one is determined when the value of Y equals/ (is less than) the value of D × (1 − Itn/MxItn) , the tactic of the silverback could be selected that can directs the others to food sources. This tactic can be represented mathematically as signified in (15) as follows: If the value of Y is more than the term D × (1 − Itn/MxItn) , the tactic of competing for adult females is selected 38 . This tactic can be represented mathematically as signified in (17) as follows: where L is the force of impact; rn 5 is random number from [0:1]; β is pre-optimization value which is specified and set to 3; The factor (A) vector is the violence level in a fight; and E is employed as imitator for the violence efficacy.
The GtX(Itn) solution will replace X(Itn) if the fitness value of GtX(Itr) is less than X(Itn).
Improved GTT incorporating tangent flight strategy. In this part, an improved version of the GTT (IGTT) incorporating Tangent Flight Strategy (TFS). The Cauchy is computed as follows, and its tangent function is the same for the TFS 44 : where pp is a uniformly distributed arbitrary number with a value in the interval [0, 1], and Dim is the number of dimensions in the function. This operation is capable of efficiently searching the search space. This function is periodic, and it does not break the balance between both exploration and exploitation. The TFS is added to Eq. (15) by the suggested IGTT approach. The gorilla and silverback's separation will narrow as a result of this modification, drastically reducing the ultimate step size and improving the objective value. This model may be explained mathematically as follows: where pp is evaluated using Eq. (19). The key steps for the proposed IGTT are illustrated as depicted in Fig. 2 36 . As shown, the five behaviors in optimizing the gorillas are highlighted in green.

Simulation results and discussion
Firstly, a comparative assessment of the developed IGTT and the original GTT are demonstrated on the ten common, and well-known benchmark mathematical models following to the Congress on Evolutionary Computation (CEC) 2017 unconstrained benchmark functions 45 . Their mathematical objective model, dimensions, ranges of the control variables and their optimal objective value are announced in Table 1. The first function (F1) represents a unimodal function while the second one (F2) is a multimodal function. The functions (F3-F6) represent mixed functions, and the functions (F7-F10) are composite functions. Table 2 shows the performance study of the developed IGTT and the original GTT for ten common, wellknown mathematical models and how it compares to two well-established, and well-known optimizers such as GWA 46 and particle swarm optimization (PSO) 47 . Also, the best regarding convergence characteristics are displayed in Fig. 3. This Table clearly shows that the IGTT performs and operates more effectively than the original GTT, PSO and GWA in the tested mathematical functions, demonstrating the robustness of IGTT in finding the best answer to these mathematical functions. From this Table, the suggested IGTT outperforms the standard GTT in 92.5% of the statistical indices of the investigated benchmark functions for the best, mean, worst, and standard deviations. Similarly, compared to the PSO, the developed IGTT outperforms it in 92.5% of the statistical indices of the investigated benchmark functions. Compared to the GWA, the IGTT outperforms it in 87.5% of the statistical indices of the investigated benchmark functions.
After that, two instances of common commercial PEMFCs stacks are discussed: the BCS 500-W and the Modular SR-12 PEM units in this study to manifest the performances of the developed IGTT to obtain parameter extraction of FC. In addition to this, different new recently inspired optimizers are implemented for comparative validation which are the original GTT, SDO, FFO and RFO. The compared techniques are performed in the MATLAB environment (MATLAB 2017b) using PC with Intel(R) Core(TM) i7-3632QM CPU @ 2.20 GHz and 8 GB RAM. For fair comparisons, similar circumstances are taken into considerations with 50 solutions as a population size and 100 iterations as a maximum number. It is commonly known that meta-heuristics have a high level of randomness. As a result, the exhibited minimal SSE results are obtained after 100 separate executions, as  54 . Also, the GTT shows a significant supremacy compared to ALO 27 , MFO 51 and VSDE 54 which obtain SSE objectives of 0.0119, 0.0119 and 0.01214, respectively. Additionally, the developed IGTT shows a small comparable preponderance compared to SMS 2 , IHBO 49 and FMHHO 53 which obtain SSE objectives of 0.0169778, 0.0117 and 0.01177, respectively.
To contrast the robustness validation of the developed IGTT with GTT, SDO, FFO and RFO, Fig. 4 describes the best SSE values of a 30 run times sample. As shown, the relative optimum SSE values are related to the developed IGTT where the attained results by the developed IGTT always supersede the GTT, SDO, FFO and RFO as represented in that figure. Not only that, but Table 5 illustrates their comparative assessment for the BCS 500W Stack with several other published results through the best, mean, worst and STD over the separate runs. As shown, the proposed IGTT has the best effectiveness since it acquires the least good, mean, worst and STD values of 0.011697781, 0.014329, 0.02699 and 0.0053594, respectively. Also, Fig. 5 depicts the best convergence curves related to the IGTT, GTT, SDO, FFO and RFO for BCS 500W Stack. As shown, the IGTT has the fastest response in finding the minimum SSE in approximately 30% of the iteration's axis.
Based on the IGTT parameters extraction for BCS 500W Stack, Fig. 6 shows the regarding I/V and P/V characteristics compared to the related experimental recordings (The regarding values are tabulated in the appendix, kindly refer to Table A.1). As shown, excellent fittings among the simulated and measured I/V and P/V characteristics are observed.
In contrast, the described polarization characteristics in terms of I/V and I/P plots are shown in Fig. 7a-c. First, the I/V curves are plotted under the pressures of P H 2 /P O 2 of 1.000/0.2095 bar, 1.5/1.0 bar, and 2.5/1.5 bar; respectively, at a constant cell temperature of 333 K which are shown in Fig. 7a. Then, the temperature's variations are simulated at 303 K, 333 K and 373 K; respectively at constant partial pressures as specified in the datasheet (i.e. P H 2 /P O 2 =1.0/0.2095) which are depicted in Fig. 7b. In addition to that, the I/P curves are plotted under varied temperatures at 60 °C, 70 °C, and 80 °C, respectively as depicted in Fig. 7c. These curves are exceptionally smooth under various operating situations, offering confidence in the IGTT-based model's high efficiency.
Test case 2: modular SR-12. The parameter extraction algorithms are thoroughly validated using the Modular SR-12 PEMFCs to verify how well the IGTT based-parameter extraction approach performs. The IGTT, GTT, SDO, FFO and RFO are performed to obtain the best parameter extraction for this model where their best obtained values and regarding SSE objective are tabulated in Table 6. In addition to this, different published outcomes of recently inspired optimizers are added in this table such as WOA 18 , flower pollination algorithm (FPA) 30 55 .
As demonstrated, when compared to SDO, FFO, and RFO, the IGTT besides GTT have the capability to achieve the best performance with the smallest SSE target. Additionally, the formed IGTT claims the best performance with the least SSE when compared to published findings. The IGTT with GTT, SDO, FFO, and RFO's best SSE values from a sample of 30 runs are shown in Fig. 8 for the robustness comparison. The relative optimum SSE values are related to the IGTT, where the IGTT's obtained results always take precedence over the GTT, SDO, FFO, and RFO given in that figure. Additionally, Table 7 compares their evaluation of the Modular SR-12 Stack with several other published results using the best, mean, worst, and STD across many runs. As evident, Table 1. Detailed definition of the ten common, well-known mathematical models under consideration. Additionally, Fig. 9 shows the IGTT, GTT, SDO, FFO, and RFO for Modular SR-12 stack's finest convergence properties. As demonstrated, the zooming part in this figure is dedicated for illustrating the capability of the IGTT and GTT in faster reaching the optimal solution after only 35% of the iteration journey while the SDO approaches to a very close value after 92% of the journey. On the other side, RFO and FFO fails to achieve a close value through 100% of the iteration journey.
Interestingly, Fig. 10 depicts the relevant I/V and P/V curves in comparison to the relevant experimental recordings based on the IGTT's parameters extraction for Modular SR-12 Stack. The generated and measured I/V and P/V curves provide excellent fits, as seen. Confirmation of this, Fig. 11 displays the regarding absolute errors between the experimental and the simulated curves (The regarding values are tabulated in the appendix, see Table A.2). It can be noticed that the maximum error of the voltage data points is lesser than 0.016% while the maximum error of the power data points is lesser than 0.015%. Table 2. Performance study of the IGTT, GTT, PSO and GWA for ten common, well-known mathematical models. 1 indicates the advantage of the proposed IGTT while 0 indicates equality or disadvantage.

Conclusion
In this article, an advanced IGTT incorporating a Tangent Flight Strategy (TFS) within the exploitation stage has been employed to effectively extract the PEMFC model's unidentified parameters. A precise model of the PEMFCs is created via the IGTT that delivers accurate simulation and modelling results for two industrial FCs with type of BCS 500W and Modular SR-12 Stacks. Through the IGTT development, the output voltage model of FC's total squared error between the measured and its optimally estimated is minimized for both PEMFCs   Table 4. Extracted parameters using the proposed IGTT, recent, and reported optimizers for the BCS 500W stack.    www.nature.com/scientificreports/ Based on the successful application of the IGTT for PEMFC modules parameter estimation in this paper, as a future research trend, the IGTT algorithm is recommended to be employed to solve further advanced engineering problems especially in power systems such as controllers design for power system stability including renewable sources, battery models identification, optimal operation of power systems with renewable sources penetrations.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.